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## The master equation and relaxation to equilibrium

We already saw that every CPTP map can be decomposed in terms of Kraus operators. Namely is a CPTP map if and only if ( sys:) = Pj Mj sys:Mj with fMjg a family of operators on the system Hilbert space such that Pj Mj Mj = Isys:. Similarly if the reduced evolution in continuous time is Markovian, then it is given by a continuous semi{group of CPTP maps whose generator L has a characteristic expression. Namely if for any t s and any sys:, t( sys:) = t s( s( sys:)) then t( sys:) = etL sys: (2.37) with L( sys:) = i[H; sys:] + Xj Lj sys:Lj 1 fLj Lj; sys:g (2.38) . where fLjg is a family of operators on the system, H is an Hermitian system operator and fA; Bg = AB + BA. This expression can be separated into two parts. The unitary like part i[H; sys:] would correspond to a unitary evolution of the system if the second part is null. j j 2 f j g.

The second part, Lj sys:L 1 L Lj; sys: , is some times called the dissipative part. It is responsible for the non reversible evolution. Note that this decomposition is not unique. If P we change Lj ! Lj + cjIsys: (2.39) i H!H+ Xj( cj Lj cjLj ) (2.40) 2 L is not modi ed. Similarly let u be a unitary matrix then the modi cation Lj ! k ujkLk generator.

does not modify L. Hence we have di erent Lj’s and H giving the same semi{group P In particular, we can always choose the Lj and H such that all the Lj’s have null trace. The operator L is called a Lindblad operator or Lindbladian. It is named after one of the researchers who established the general form of CP map semi{group generators [42,59,73]. A derivation of the expression of L starting from the semi{group property can be found in [36].

Note that actually the characterization of CP map semi{groups through the expression of their generator is not restricted to trace preserving maps. In general a semi{group of maps acting on the space of operators acting on the system Hilbert space, 0t = etL0 , is a semi{group of CP maps if and only if it exists a CP map and a system operator K such that L 0 (X)=K X+XK+ (X) (2.41).

If moreover K + K + (Isys:) = 0, the maps are unital. Namely t0 (Isys:) = Isys: for any t. We say the semi{group etL0 is unital. Therefore the dual maps t0 are trace preserving. Note that the expression we gave above for L imply directly the trace conservation. We L have tr[ (Xsys:)] = 0 for any system operator Xsys:. We say the CP semi{group etL is trace preserving if every map etL is trace preserving. In the physics literature the reduced system evolution di erential equation d sys:(t)=dt = L( sys:) (2.42) is often called the master equation in reference to its classical counter part. From the study of we can study L spectrum and therefore the large time behavior of etL sys:. First since etL is trace preserving, the spectrum of L is in the non positive real part half plane of the complex plane. L has at least one eigenvector inv: with eigenvalue 0.

If et0L is irreducible then etL is irreducible for any t [103]. So we say the semigroup is irreducible. If etL is irreducible, it accepts a unique positive de nite invariant state inv: > 0 such that L( inv:) = 0. Moreover for CPTP semi{groups, irreduciblity implies that the unique invariant state is attractive [103]. Thus for any initial system state limt!1 etL sys: = inv:. The convergence is exponential with a rate depending on the spectrum of L [65]. Let 0 = maxfRe( )= 2 spec(L) n f0gg < 0. Then for any > 0, etL sys: = inv: + O(e ( 0 )t): (2.43).

### Reduced Markovian approximations

Let us now motivate the Markovian description of open quantum systems with the presenta-tion of some physical limits leading to memoryless reduced dynamics. Hence we present some limits where G(t) ! (t) and G0(t) ! (t). For a more comprehensive presentation of Marko-vian limit of the reduced evolution, we refer the reader to the review of H. Sophn [92]. The only limit not contained in this review is the continuous time limit of repeated interactions.

#### Singular limit

Let us rst introduce the singular limit. It is a good rst example of Markovian limit. The derivation of this limit is due to P. F. Palmer [81]. For this limit, let us keep the scaling in Htot: we introduced in last section. Htot: = 2Hsys: + Henv: + (D a (g) + D a(g)): (2.44).

We are interested in the limit ! 0. In this limit, the system energy becomes much smaller than all environment energy scales. The scaling of the interaction implies that the typical relaxation rate is of the same order as the system energy scale. Hence the typical system evolution time is of the same order as the relaxation time due to the environment. The typical environment evolution time is much smaller. Hence we expect that the intrinsic environment evolution will drive it. Thus it should rapidly forgot the e ect of the interaction with the system.

The typical system evolution time is modi ed from t to t 2. We are interested in the system evolution so we scale time such that the new time is coherent with the system typical evolution time. This time scale as the inverse of the energy, thus as 2. Hence the new time scale tnew 2 = tinit: is convenient. Under this new time scale, the Hamiltonian becomes Htot: = Hsys: + 2Henv: + 1(D a(g) + D a (g)): (2.45).

This Hamiltonian scaling is more common in the literature on singular coupling than the previous one. It is actually in this scaling that singular coupling is meant. At rst glance it could be interpreted as a strong coupling limit. But as we saw it is more a weak coupling limit where the system energy is also considered small. Hence the environment should drive the evolution at short times. This is what is meant by the scaling of the environment Hamiltonian. Computing the two time correlation functions with this scaling we obtain a scaling 2G(s 2) and 2G0(s 2). If we assume that G and G0 are integrable over R+, then lim 2G( s 2) = C (s) (2.46) 0 j j.

**Weak coupling limit**

The limit we present now has become a standard example of a situation where Markovian limit is relevant. It is also know as the rotating wave Born{Markov approximation.

This limit was rst shown by E.B. Davies [46]. Actually the singular coupling limit was obtained as a modi cation of the weak coupling limit proof. Complete derivations of the rotating wave Born{Markov approximation can be found in [36, 57]. Contrary to the previous limit, here, no scaling is made on the system Hamiltonian. The only assumption is that the interaction with the environment is weak and that the environment behave well. Namely, the two time correlation functions of the environment must decay rapidly enough as the time interval grows. Hence, the memory of the environment should be short enough. This translates into the same assumption as before for the two time correlation functions. We assume it exists > 0 such that G(0)(s) < 1 and G(0)(s)s < 1.

**Repeated interactions continuous limit**

As explained earlier another good open system model is the repeated interaction one. By construction, in discrete time, it is Markovian. Knowing the system state at time k, the system state state at time n k is simply sys:(n) = n k( sys:(k)).

As pointed out by S. Attal and Y. Pautrat [6], this model should also be a good discrete time approximation of continuous time Markovian open system evolutions. They actually showed that, in the continuous time limit, with an adapted scaling, the repeated interaction model (without tracing out the probes degree of freedom) leads to a quantum stochastic evolution. We will discuss this limit in next chapter. For now let us sketch how the continuous time limit of the reduced system dynamic should lead to a CPTP semi{group evolution.

Assume the interaction happen during a time 1=n. Actually this 1=n is simply a scaling of the interaction time. It is an adimensional quantity. The time factor needed to obtain the actual time is included in the de nition of the operators involved in the Hamiltonian. Hence the eigenvalues of each part of the Hamiltonian are also adimensional. In this whole thesis we will often consider such adimensional time.

**Table of contents :**

**1. Introduction **

**2. Open system Markovian description **

2.1. Open systems

2.1.1. System evolution and completely positive maps

2.1.2. Repeated interactions

2.1.3. Caldeira{Leggett model

2.1.4. Path integral approach

2.2. Markovian evolution

2.2.1. Memory issue

2.2.2. The master equation and relaxation to equilibrium

2.3. Reduced Markovian approximations

2.3.1. Singular limit

2.3.2. Weak coupling limit

2.3.3. Repeated interactions continuous limit

2.3.4. Low density limit

**3. Markovian environment description { Quantum noises **

3.1. Quantum noises

3.2. Open systems extended Markovian limits

3.2.1. Master equation dilation

3.2.2. Weak coupling

3.2.3. Repeated interactions continuous limit

3.3. Corresponding path integral description and classical limit

3.3.1. Path integral for an environment in its vacuum state

3.3.2. Out of vacuum environments

3.3.3. Classical limit

**4. Repeated indirect measurements **

4.1. Discrete time quantum trajectories

4.1.1. Indirect measurement

4.1.2. Repeated indirect measurements

4.2. Asymptotic behavior

4.2.1. Ergodicity

4.2.2. State purication

4.3. Non demolition indirect measurements and wave function collapse

4.3.1. Introduction to non demolition measurements

4.3.2. Non demolition measurement and martingale change of measure .

4.3.3. A note on degenerate measurements

4.4. Invariant subspace stabilization

4.4.1. Invariant and globally asymptotically stable subspaces

4.4.2. Invariance and asymptotic stability in mean

4.4.3. Almost sure invariance and asymptotic stability

4.4.4. Exponential convergence

**5. Continuous indirect measurements **

5.1. Continuous time quantum trajectories

5.2. Physical models of continuous indirect measurement

5.2.1. Phenomenological approach

5.2.2. Repeated indirect measurement continuous approximation

5.2.3. Quantum ltering

5.2.4. Unraveling of master equation

5.3. Asymptotic behavior

5.3.1. Ergodicity

5.3.2. State purication

5.4. Non demolition indirect measurements and wave function collapse

5.4.1. Non demolition condition

5.4.2. Wave function collapse

5.4.3. Exponential convergence rate

5.4.4. Mean convergence time

5.4.5. Collapse stopping time

5.4.6. Estimated state stability

5.5. Invariant subspace stabilization

5.5.1. Invariant and globally asymptotically stable subspaces

5.5.2. Invariance and asymptotic stability in mean

5.5.3. Almost sure invariance and asymptotic stability

5.5.4. Exponential convergence

**6. Conclusion **

**A. Mean convergence time proofs **

A.1. QND nite mean convergence time

A.2. Asymptotic subspace nite mean converge time

A.3. Continuous QND nite mean convergence time

A.4. Continuous asymptotic subspace nite mean convergence time

**B. Path integral derivations **

B.1. Derivation of the path integral for an environment in its vacuum

B.2. Path integral derivation for nt 6= 0

B.3. Classical limit derivation

**C. Article: Repeated quantum non-demolition measurements: convergence and continuous- time limit **

C.1. Introduction

C.2. QND measurements as stochastic processes

C.2.1. Repeated indirect quantum measurements

C.2.2. A toy model

C.3. Measurement apparatus and Bayes’ law

C.4. Convergence

C.4.1. Convergence with dierent partial measurement methods

C.4.2. Conditioning or projecting

C.4.3. Convergence rates and trial distribution independence

C.4.4. Convergence rate tuning

C.5. Degeneracy and limit quantum state

C.5.1. State distribution convergence

C.5.2. Density matrix convergence

C.6. Continuous diusive limit

C.6.1. Continuous time limit of the pointer state distribution

C.6.2. Density matrix evolution

C.A. Details for mutual singularity

C.B. Proof of existence of a continuous time limit

C.C. Details on the continuous time limit with dierent partial measurement methods194

C.D. Derivation of the density matrix evolution

**D. Article: Iterated Stochastic Measurements **

D.1. Introduction

D.2. Iterated indirect stochastic measurements

D.2.1. Discrete time description

D.2.2. Continuous time limit

D.3. Iterated indirect quantum measurements

D.3.1. Discrete time description

D.3.2. Continuous time limit

**E. Article: Large Time Behaviour and Convergence Rate for Non Demolition Quantum Trajectories **

E.1. Non destructive quantum trajectories

E.1.1. System state evolution

E.1.2. Non demolition condition

E.2. Convergence and wave function collapse

E.2.1. Wave function collapse

E.2.2. Exponential rate of convergence

E.2.3. Stability